Hyperpent and hypopent

Hyperpent or hyperpental edo is an equal division of the octave which has a just fifth that is less than a syntonic comma sharp. It is juxtaposed by hypopent or hypopental edo tuning systems.

Background

"Hyper-" is the Greek prefix for high or above and "pent" is Greek for five. The term is loosely associated with "superpythagorean," but specifically refers to equal division of the octave that result in any fifth tempered sharp.

Large edo's are generally both hyperpent and hypopent, because they contain multiple representations of the fifth that may be sharp and flat. Some edo's may contain no intervals in the span of a just fifth plus or minus a syntonic comma, and thus are neither hyperpent nor hypopent.

Partial List

Anpent (neither hypopent nor hyperpent)

2edo 3edo 4edo 6edo 8edo 9edo 11edo 13edo 16edo 18edo 23edo

Hyperpent

5edo 10edo 15edo 17edo 20edo 22edo 25edo 27edo 29edo 30edo 32edo 34edo 39edo 41edo 46edo

Hypopent

7edo 12edo 14edo 19edo 21edo 24edo 26edo 28edo 31edo 33edo 36edo 38edo 43edo

Amphipent (both hyperpent and hypopent)

35edo 37edo 40edo 42edo 44edo 45edo 47edo

Analysis

If 5edo and 7edo are taken as the smallest hyperpent and hypopent edo's, respectively, other edo tuning systems (let's call it "Xedo" with "X" equal divisions of the octave) can be determined to be hyperpent, hypopen, anpent, or amphipent by addition as follows (with a few exceptions):

If X can be represented as a sum of 5's, Xedo is hyperpent. If X can be represented as a sum of 7's, Xedo is hypopent. If X can be represented as a sum of 5's and 7's, with more 5's than 7's, Xedo is hyperpent. If X can be represented as a sum of 5's and 7's, with more 7's than 5's, or an equal number of 7's and 5's, Xedo is hypopent. If X cannot be represented as a sum of 5's and 7's, Xedo is anpent. If X can be represented as a sum of 5's and 7's in multiple ways (for example, 35 can be represented as five sevens or as seven fives), Xedo is amphipent.

Some exceptions are 37edo and 44edo, which are amphipent, even though 37 cannot be written as multiple sums of 5's and 7's, nor can 44.